Unit 7 MC Review

A family owns five vehicles with the following durations of ownership: 2, 4, 6, 10 and 12 years. Suppose you select a random sample of two of the vehicles and calculate the mean duration of ownership. Which of the following dotplots shows the sampling distribution of the sample mean?

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Two different formulas are being used to estimate the standard deviation of a population using data from a sample. Using technology, 100 random samples of size n are selected from the population. For each random sample, the two estimates were computed. The histograms display the simulated sampling distributions of the estimators. The population standard deviation is 1. Which of the following is true?

• Estimator A and Estimator B both have low bias, but Estimator A has lower variability then Estimator B.

• Estimator A has lower bias and higher variability than Estimator B.

• Estimator A and Estimator B both have low bias and similar variability.

• Estimator A and Estimator B both have high bias and similar variability.

• Estimator A and Estimator B both have high bias, but Estimator A has lower variability then Estimator B.

Answer explanation

Both estimators have low bias, indicating they are accurate on average. However, Estimator A shows lower variability, meaning its estimates are more consistent compared to Estimator B, making the first answer choice correct.

Vermilion flycatchers may adjust their song lengths in noisy conditions. Researchers in Mexico City will use data from a simple random sample of vermilion flycatcher songs to estimate mean song length (in seconds) in high-traffic areas. Which of the following statements is correct about the sample mean as a point estimator?

• A sample size of 50 will produce more variability in the estimator than a sample size of 100.

• A sample size of 50 will produce less variability in the estimator than a sample size of 100.

• A sample size of 50 will produce more bias in the estimator than a sample size of 100.

• A sample size of 50 will produce less bias in the estimator than a sample size of 100.

• A sample size of 50 will produce more variability and less bias in the estimator than a sample size of 100.

Answer explanation

A sample size of 50 will produce more variability in the estimator than a sample size of 100 because larger samples tend to provide more stable estimates, reducing variability in the sample mean.

Extreme heat is the leading cause of weather-related death in the United States. A recent survey in Texas found that 20% of respondents reported experiencing heat-related symptoms such as dizziness, fatigue and headaches. Assume that the true population proportion of Texas residents who experience heat-related symptoms is 0.20. If a random sample of 56 Texas residents is selected, which of the following describes the sampling distribution of p^​, the sample proportion of Texas residents who experience heat-related symptoms.

• The sampling distribution is skewed left, with mean of 0.20 and standard deviation of 0.053.

• The sampling distribution is skewed right, with a mean of 0.80 and standard deviation of 0.400.

• The sampling distribution is approximately normal with mean of 0.20 and standard deviation of 0.053.

• The sampling distribution is approximately normal with a mean of 0.80 and standard deviation of 0.053.

• The sampling distribution is approximately normal with mean of 0.20 and standard deviation of 0.400.

Answer explanation

The sampling distribution of p^ is approximately normal due to the sample size (n=56) being large enough. The mean is the population proportion (0.20), and the standard deviation is calculated as sqrt[(0.20)(0.80)/56] = 0.053.

Based on many years of sales data, a small retail store knows that the distribution of their daily sales is skewed right with a mean of $783 and a standard deviation of $106. Suppose a random sample of 36 days selected and the sample mean daily sales is calculated. This process is repeated for a total of 150 samples. Which of the following is the best description of the distribution of the 150 sample means?

• Skewed right with a mean of $783 and a standard deviation of $106

• Skewed right with a mean of $783 and a standard deviation of $17.67

• Skewed right with a mean of $783 and a standard deviation of $8.65

• Approximately normal with a mean of $783 and a standard deviation of $17.67

• Approximately normal with a mean of $783 and a standard deviation of $8.65

Answer explanation

The distribution of sample means will be approximately normal due to the Central Limit Theorem, with a mean of $783. The standard deviation of the sample means is $106/√36 = $17.67.

The distribution of annual medical expenses per household is strongly skewed right. If random samples of size 30 are selected from the population distribution, which of the following statements about the sampling distribution of the sample mean medical expenses per household is true?

• The mean and the standard deviation of the sampling distribution are approximately the same as the mean and the standard deviation of the population distribution.

• The shapes of the sampling distribution and the population distribution are the same because the sample size is 30.

• If the sample size were increased, the standard deviation of the sampling distribution would get closer to the standard deviation of the population distribution.

• If the sample size were increased, the shape of the sampling distribution would get closer to a normal distribution.

• Because the sample size is 30, the population is approximately normal.

Answer explanation

The Central Limit Theorem states that as sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's shape. Thus, the correct choice is that increasing the sample size will make the shape closer to normal.

The distribution of lengths of time that homeowners have owned their current homes in a certain town is approximately normal with a mean of 12.9 years and a standard deviation of 4.1 years. Nine homeowners are randomly selected from this population. What is the probability that the average length of time that these nine homeowners have owned their current house is greater than 14 years?

• 0.0079

• 0.2104

• 0.3942

• 0.7896

• Essentially 0

Answer explanation

To find the probability that the average is greater than 14 years, normcdf. The mean stays the same and the standard deviation we use the SD formula for sample means, SD/sqrt(n). We then plug in normcdf(14, 9999, 12.9, 1.3666). The corresponding probability is 0.2104.

In a long-term study conducted in Poland, it was found that the distribution of birth weights for full-term male babies has a mean of 3622 grams and a standard deviation of 432 grams. The distribution of birth weights for full-term female babies has a mean of 3465 grams and a standard deviation of 414 grams. Suppose a researcher selects independent random samples of 6 full-term male babies and 8 full-term female babies. Which of the following expressions gives the standard deviation of the sampling distribution of the difference in mean weights?

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